Abstract Algebra/Modules

MotivationEdit

Let G be an abelian group under addition. We can define a sort of multiplication on G by elements of   by writing   for   and  . We can extend this to the case where n is negative by writing  . We would, however, like to be able to define a sort of multiplication of a group by an arbitrary ring.

DefinitionEdit

Definition 1 (Module)
Let R be a ring and M an abelian group. We call M a left R-module if there is a function  , called a scalar multiplication, satisfying
  1.  ,
  2.  , and
  3.  
for all  .
We call R the ring of scalars of M.

Note: We can also define a right R-module analogously by using a function  . In particular the third property then reads:

 

Note that the two notions coincide if R is a commutative ring, and in this case we can simply say that M is an R-module.

Definition 2: Given any ring R, we can define it's opposite ring,  , having the same elements and addition operation as R, but opposite multiplication. Their multiplication rules are related by  . In contrast to group theory, there is no reason in general for a ring to be isomorphic to its opposite ring.

The obervant reader will have noticed that the scalar multiplication in a left R-module M is simply a ring homomorphism   such that   for all  . We leave it as an exercise to verify that the scalar multiplication in a right R-module is a ring homomorphism  . Thus a right R-module is simply a left Rop-module. As a consequence of this, all the results we will formulate for left R-modules are automatically true for right R-modules as well. There are no assumptions that the module is unital, namely that 1m = m for all m in M.

Examples of ModulesEdit

  1. Any ring R is trivially an R-module over itself. More interestingly, any left ideal I of R is also a left R-module with the obvious scalar multiplication. In addition, if I is a two-sided ideal of R, then the quotient ring   is an R-module with the induced scalar multiplication  .
  2. If R is a ring, then the set   of   matrices with entries in R is an R-module under componentwise addition and scalar multiplication. More generally, for any set X, the set   of function from X to R, with or without finite support, is an R-module in an obvious way.
  3. The k-modules over a field k are simply the k-vector spaces.
  4. As was shown in the introduction of this chapter, any abelian group is a  -module in a natural way. ("Natural" here has a rigorous mathematical meaning which will be explained later.
  5. Let S be a subring of a ring R. Then R is an S-module in a natural way. We can extend this as follows. Let S,R be rings and   a ring homomorphism. Then R is an S-module with scalar multiplication   and   for all  .
  6. Any matrix ring of a ring R is a R-module under componentwise scalar multiplication.
  7. If S is a subring of a ring R, then any left R-module is also a left S-module with the restricted scalar multiplication. We will treat this more generally later.

SubmodulesEdit

Definition 3: (Submodule)

Given a left  -module   a submodule of   is a subset   satisfying
  1. N is a subgroup of M, and
  2. for all   and all   we have  .

The second condition above states that submodules are closed under left multiplication by elements of  ; it is implicit that they inherit their multiplication from their containing module;   must be the restriction of  .

Example 4: Any module M is a submodule of itself, called the improper submodule, and the zero submodule consisting only of the additive identity of M, called the trivial submodule.

Example 5: A left ideal I is a submodule of R viewed as an S-module, where S is any (not necessarily proper) subring of R.

Lemma 6: Let M be a left R-module. Then the following are equivalent.

i) N is a submodule of M
ii) If   and   for all  , then  .
iii) If   and  , then  .

Proof: i) => iii):   and   are in   by the second property, then   by the first property of Definition 3.

iii) => ii): Follows by induction on  .

ii) => i): Let  ,  , then for arbitrary   be have  , proving   is a subgroup. Now let  , then for arbitrary  ,  , proving property 2 in Definition 3.

The lemma gives an alternative characterisation of submodules, and those sets closed under linear combinations of elements.

Analogously to the case of vector spaces, we have ways of creating new subspaces from old ones. The rest of this subsection will be concerned with this.

Lemma 7: Let M be a left R-module, and let N and L be submodules of M. Then   is a submodule in M, and it is the largest submodule contained in both N in L.

Proof: Let   and  . Then   and   since N and L are submodules, so   and   is a submodule of M. Now, assume that S is a submodule of M contained in N and L. Then any   must be in both N and L and therefore in   such that  , proving the lemma.

Now, as the reader should expect at this point, given submodules N and L of M, the union   is in general not a submodule. In fact, we have the following lemma:

Lemma 8: Let M be a left R-module and let N and L be submodules. Then   is a submodule if and only if   or  .

Proof: The left implication is obvious. For the right implication, assume   is a submodule of M. Then if   and  , then  , which implies that   or  . Assume without loss of generality that  . Then, since N is a submodule, we must have  , proving  .

Definition 9: Let M be a left R-module, and let   be submodules for  . Then define their sum,  .

Definition 9 has a straightforward extension to sums over arbitrary index sets. This definition is left for the reader to state. We will only need the finite case in this chapter.

Lemma 10: Let M be a left R-module and let N and L be submodules. Then   is a submodule of M, and it is the smallest submodule containing both N and L.

Proof: It is straightforward to see that   is a submodule. To see that it is the smallest submodule containing both N and L, let S be a submodule containing both N and L. Then for any   and  , we must have  . But this is the same as saying that  , proving the lemma.

With Lemma 7 and Lemma 10 established, we can state the main result of this subsection.

Definition 11: Let M be a left R-module. Then let   be the set of submodules ordered by set inclusion.

Lemma 12: Let M be a left R-module. Then   forms a lattice, the join of   being given by   and their meet by  .

Proof: Most of the work is already done. All that remains is to check assosiativity, the absorption axioms and the idempotency axioms. The associativity is trivially satisfied,   and   for all  . As for absorption, We have to check   and   for all  , but this is also trivially true. Lastly, we obviously have   and   for all  , so we are done.

Corollary 13: Let M be a left R-module. Then   is a modular lattice.

Note: Recall that   is modular if and only if whenever   such that  , we have  .

Proof: Let   such that  . Since  , we have   for some  , such that  . Thus   and  . On the other hand, we have   and  , so  .

Definition 14: Let M be a left R-module. A submodule N is called maximal if whenever L is a submodule satisfying  , then   or  .

Theorem 15: Every submodule of a finitely generated left R-module is contained in a maximal submodule.

Proof: Let N be a submodule, and let  . Then S is a poset under set inclusion. Let   be a chain in S, and note that   is a submodule containing each  , such that U is an upper bound for the chain. Then, since each chain in S has an upper bound, by Zorn's Lemma S has a maximal element, P, say. P is obviously an ideal containing N. By the definition of S, P is also a maximal submodule of M, proving the theorem.

Generating ModulesEdit

Given a subset   of a left  -module  , we define the left submodule generated by   to be the smallest submodule (w.r.t. set containment) of   that contains  . It is denoted by   for a reason which will become clear in a moment.

The existence of such a submodule comes from the fact that an intersection of  -modules is again an  -module: Consider the set   of all submodules of   containing  . Since   contains  , we see that   is non-empty. The intersection of the modules in   clearly contains   and is a submodule of  . Further, any submodule of   containing   also contains the intersection. Thus  .

Assuming that   is unitary, the elements of   have a simple description;

 .

That is, every element of   can be written as a finite left linear combination of elements of  . This equality can be justified by double inclusion: First, any submodule containing   must contain all left  -linear combinations of elements of   since modules are closed under addition and left multiplication by elements of  . Thus,  . Secondly, the set of all such linear combinations forms a submodule of   containing   (use   and  ) and hence it contains  .

Generating Submodules by IdealsEdit

Consider any ring  , left ideal  , and left  -module  . One can think of   as a subring of   (non-unitary when  ) and hence   is an  -module using the regular multiplication by elements of  .

If we consider the set   we obtain a submodule of  . This follows from our discussion of generated submodules. However, since   is not unitary, it is not necessary that  .

Thus, we may consider the quotient module  . Clearly this is an  -module but it is also an   module under the obvious action.

Proposition
Given an  -module   and ideal   of  , the  module   is an  -module with multiplication  .
proof.
To show that this is well defined, we observe that if   then   and hence
 
since  . Thus,
 
which proves that the action of   on   is well defined. It follows now that   is an  -module simply because it is an  -module.

Quotient ModulesEdit

Recall that any subgroup   of an abelian group   allows one to construct an equivalence relation; for  ,

 .

Cosets of  , equivalence classes under the relation above, can then be endowed with a group structure, derived from the original group, and is given the name M/N. The sum of two cosets   and   is simply  .

Lemma 16 Let M be a left R-module and N be a submodule. Then M/N, defined above, is a left R-module.

Proof: M/N is obviously an abelian group, so we just have to check that it has a well-defined R-action. Let   and  . Then we define  . The distributivity and associativity properties of the action are inherited from M, so we just need well-definedness. Let   with  . Then   since N is a submodule, and we are done.

Module HomomorphismsEdit

Like all algebraic structures, we can define maps between modules that preserve their algebraic operations.

Definition (Module Homomorphism)
An  -module homomorphism   is a function from   to   satisfying
  1.   (it is a group homomorphism), and
  2.  

When a map between two algebraic structures satisfies these two properties then it called an  -linear map.

Definition (Kernel, Image)
Given a module homomorphism   the kernel of   is the set
 
and the image of   is the set
 .

The kernel of   is the set of elements in the domain that are sent to zero by  . In fact, the kernel of any module homomorphism is a submodule of  . It is clearly a subgroup, from group theory, and it is also closed under multiplication by elements of  :   for  .

Similarly, one can show that the image of   is a submodule of  .